Methods and systems for determining optical propertis using low coherence interference signals

ABSTRACT

Methods and related systems for determining properties of optical systems (e.g., interferometers) and/or optical elements (e.g., lenses and/or lens systems) are described. For example, information related to an optical thickness mismatch of an interferometer can be determined by providing scanning interferometry data. The data typically include obtaining one or more interference signals each corresponding to a different spatial location of a test object. A phase is determined for each of multiple frequencies of each interference signal. The information related to the optical thickness mismatch is determined based on the phase for each of the multiple frequencies of the interference signal(s).

CROSS REFERENCE TO RELATED APPLICATIONS

Pursuant to 35 U.S.C. § 120, this application is a continuationapplication of U.S. application Ser. No. 11/604,668, filed Nov. 27,2006, now U.S. Pat. No. 7,304,747, which is a divisional of prior U.S.application Ser. No. 11/131,649, filed May 17, 2005, now U.S. Pat. No.7,142,311, which claims the benefit of U.S. Provisional Application No.60/572,010, filed May 18, 2004. The contents of the prior applicationsare incorporated herein by reference in their entirety.

FIELD OF THE INVENTION

The present invention is related to methods for determining propertiesof optical elements and/or optical systems as well as to relatedsystems.

BACKGROUND

Scanning White Light Interferometry (SWLI), also known as verticalscanning or coherence radar, takes advantage of the fringe localizationin low-coherence interferometers to profile complex surface shapes.Combining phase and coherence information for improved precisionrequires dedicated algorithms for dealing with possible inconsistenciesbetween these two measurement techniques. See, for example,“Determination of Fringe Order in White-Light Interference Microscopy”,Peter de Groot, Xavier Colonna de Lega, Jim Kramer, Michael Turzhitsky,Applied Optics, Volume 41, Issue 22, 4571-4578, August 2002. Some ofthese inconsistencies are due to geometric and chromatic aberrationsthat are typical of optical components. Those same aberrations are alsoresponsible for reducing the effective lateral resolution of theseprofilers.

SUMMARY

Methods and systems are disclosed for an aberration characterizationtechnique based on the frequency domain analysis of the interferogramsgenerated by low-coherence profilers. The outputs of this analysis arefield-dependent maps of the magnitude and gradient of the variouslateral aberrations that affect the imaging system. This information canbe used to select optimum components, detect misalignments orexcentrations, and generally offer objective means of assessing thepotential lateral resolution of an instrument.

In preferred embodiments, the analysis in the frequency domain oflow-coherence interference signals extracts quantitative informationabout optical characteristics of an interferometer and its components.This information can for example be used as a feedback for an alignmentprocedure or to test the quality of optical subassemblies.

A first aspect is a procedure where the non-linearity of the phase ofthe frequency spectrum is used to calculate an optical mismatchcharacteristic of an interferometer. Given the optical properties(refractive index as a function of wavelength) of compensatingdispersive materials present in the interferometer it is then possibleto calculate a thickness correction for each material in order to obtainan interferogram having optimum contrast and phase properties.Alternatively, an iterative procedure is used to adjust the thickness ofa compensating dispersive material having unknown optical properties.

Another aspect is a procedure where the non-linearity of the phase ofthe frequency spectrum is calculated across the entire field of view ofthe interferometer. The shape of the resulting non-linearity mapprovides information about the misalignment of optical componentslocated inside the interferometer cavity. For example, in a Linnikinterferometer, decentration of the optics results to first order in atilt of the non-linearity map. A measure of the non-linearity gradientprovides quantitative feedback for alignment.

Another aspect is a procedure where the phase and/or magnitude of two ormore frequency spectrum components are used to create images of apatterned object. Image correlation techniques are then applied tomeasure the relative lateral displacement of image features, thusproviding a measure of lateral imaging aberrations of theinterferometer's imaging system. In one embodiment the technique is usedto align optical components of the interferometer. In another embodimentthe interferometer (for example of the Twyman-Green or Linnik type) isused to characterize optical components and assemblies.

In another aspect of the invention, a method includes providing scanninginterferometry data from an interferometer. The interferometer typicallyincludes multiple optical elements configured to reflect light from atest object that is different from the optical elements to determineinformation about the test object. The scanning interferometry datatypically includes an interference signal including an interferenceintensity value for each of multiple scan positions of theinterferometer. The method also includes determining a relationshipbetween phase and frequency components of the interference signal andreducing, based on the relationship between the phase and frequencycomponents of the interference signal, an optical thickness mismatch fora field position of the interferometer corresponding to the interferencesignal, the optical thickness mismatch being between the opticalelements of a reference arm of the interferometer and the opticalelements of a test arm of the interferometer.

Reducing the optical thickness mismatch can include adding at least oneadditional optical element to an optical path of the test arm or thereference arm of the interferometer. The at least one additional opticalelement may include a first optic formed of a first optical medium and asecond optic formed of a second optical medium. Reducing the opticalthickness mismatch can include determining a thickness of the additionaloptical element based on the relationship between the phase andfrequency components of the interference signal.

Reducing the optical thickness mismatch can include changing a positionof at least one the optical elements of the interferometer.

Reducing the optical thickness mismatch can include replacing one of theoptical elements of the interferometer with another optical element.

Reducing the optical thickness mismatch can be performed iteratively.For example, after changing a position of at least one of the opticalelements of the interferometer, one or more additional interferencesignals are obtained with the optical element(s) in the changedposition. A relationship between phase and frequency components of theadditional interference signal(s) is determined and can be compared withthe relationship determined before changing the position of the opticalelement(s). The iterative process can also or alternatively be used withrespect to adding an optical element and/or replacing at least oneoptical element.

Determining the relationship between phase and frequency components ofthe interference signal can include transforming the interference signalinto a frequency domain with respect to scan values for the scanpositions.

Determining the relationship between phase and frequency components ofthe interference signal can include determining a phase for each ofmultiple frequency components of the interference signal, fitting afunction to the phase determined for each of the multiple frequencycomponents, the function having at least one fitting parameter, andreducing the optical thickness mismatch based on the fitting parameter.

The scanning interferometry data can include multiple interferencesignals where each interference signal corresponds to a different fieldposition of the interferometer and the method can include determining arelationship between phase and frequency components of each interferencesignal and reducing, based on the relationship between the phase andfrequency components of the interference signal, an optical thicknessmismatch for each of multiple field positions of the interferometer eachcorresponding to at least one of the interference signals, the opticalthickness mismatch being between the optical elements of a reference armof the interferometer and the optical elements of a test arm of theinterferometer.

Reducing the optical thickness mismatch can include adding at least oneadditional optical element to an optical path of the test arm or thereference arm of the interferometer. The at least one additional opticalelement has a thickness that varies with the field position of theinterferometer.

Reducing the optical thickness mismatch can include changing a positionof at least one optical element of the interferometer. The at least oneoptical element can have a thickness that varies with the field positionof the interferometer.

Each frequency component may correspond to a wavelength of light of thelight source.

The interference signal can include interference intensity valuesobtained over a range of scan positions and the range of scan positionsis greater than a coherence length of a light source of theinterferometer.

Another embodiment of the invention relates to a method includingproviding scanning interferometry data of a test object. The scanninginterferometry data includes an interference signal for each of multiplespatial locations of the test object. Each interference signal includesan interference intensity value for each of multiple scan positions ofthe interferometer. First information about the object is determinedbased on a frequency component of each of the interference signals.Second information about the object is determined based on a secondfrequency component of each of the interference signals. The secondfrequency component is typically not used in the determination of thefirst information about the object.

The scanning interferometry data can be obtained by a method includingpassing light through an optical element, and the method can furtherinclude determining information related to the optical element based onthe first and second information about the test object.

The information related to the optical element may be related to alateral aberration of the optical element.

Determining the first and second information can include transformingeach interference signal into a frequency domain with respect to scanvalues for the scan positions of the interferometer.

The method can include determining the first information based on atleast one of a phase and a magnitude corresponding to the firstfrequency of each transformed interference signal and determining thesecond information based on at least one of a phase and a magnitudecorresponding to the second frequency of each transformed interferencesignal.

The optical element may be positioned along an optical axis of an arm ofthe interferometer. The optical element may be a lens system includingmultiple lenses.

The method can include moving the optical element relative to an opticalaxis of the arm of the interferometer in response to the informationrelated to the lateral aberration.

The method can include deciding to replace the optical element with asecond optical element based on the information related to the lateralaberration.

The optical element may be positioned along an optical axis between anarm of the interferometer and a detector of the interferometer.

The first frequency component of each interference signal may resultfrom interference of light having a first wavelength and the secondfrequency component of each interference signal results frominterference of light have a second wavelength.

The first frequency of each interference signal may result frominterference of light that illuminates the test object with a firstangle of incidence and the second frequency of the interference signalresults from interference of light that illuminates the test object witha second, different angle of incidence.

The first information about the object may be related to a first heightprofile of the test object and the second information about the objectis related to a second height profile of the test object. The method caninclude determining a first instrument transfer function of theinterferometer based on the first height profile and determining a firstinstrument transfer function of the interferometer based on the secondheight profile.

The first information about the object may be related to a firstreflectivity profile of the test object and the second information aboutthe object may be related to a reflectivity height profile of the testobject. The method can include determining a first modulation transferfunction of the interferometer based on the first reflectivity profileand determining a first modulation transfer function of theinterferometer based on the second reflectivity profile.

Another aspect of the invention relates to a method including providingfirst image data of an object, the first image data having been obtainedby passing first light through an optical element, the first lighthaving a first central wavelength and providing second image data of theobject, the second image data having been obtained by passing secondlight through the optical element, the second light having a secondcentral wavelength different from the first central wavelength. Passingsecond light through the optical element is typically performed afterthe step of passing the first light through the optical element (e.g.,by varying a property of the light source and/or by modifying a filter).At least a portion of the first image data and at least a portion of thesecond image data are cross-correlated. Information about the opticalelement is determined based on the cross-correlation of the first andsecond image data.

The first and second image data of the object may be obtained by amethod including detecting light reflected from the object, the detectedlight having the first central wavelength and then, detecting lightreflected from the object, the detected light having the second centralwavelength.

Unless otherwise defined, all technical and scientific terms used hereinhave the same meaning as commonly understood by one of ordinary skill inthe art to which this invention belongs. All references cited herein areincorporated by reference; however, in case such references conflictwith the present disclosure, the present disclosure controls.

Other features, objects, and advantages of the invention will beapparent from the following detailed description.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 illustrates an interferometer system.

FIG. 2 illustrates an interference signal obtained with aninterferometer and corresponding to a single spatial location of a testobject.

FIG. 3 illustrates an interference signal obtained with aninterferometer and corresponding to a single spatial location of a testobject.

FIG. 4 illustrates phase vs. frequency data for the interference signalof FIG. 3.

FIG. 5 illustrates an interference signal obtained with theinterferometer used to obtain the interference signal of FIG. 3 butafter positioning a compensating optical element along an arm of theinterferometer.

FIG. 6 illustrates phase vs. frequency data for the interference signalof FIG. 3.

FIG. 7 illustrates an interference signal obtained with aninterferometer and corresponding to a single spatial location of a testobject.

FIG. 8 illustrates phase vs. frequency data for the interference signalof FIG. 7.

FIG. 9 illustrates an interference signal obtained with theinterferometer used to obtain the interference signal of FIG. 7 butafter positioning a compensating optical element formed of a singleoptical medium along an arm of the interferometer.

FIG. 10 illustrates residuals related to the phase vs. frequency data ofFIG. 8.

FIG. 11 illustrates an interferometer having a compensating opticalelement that can reduce dispersion mismatch.

FIG. 12 illustrates an interferometer having a compensating opticalelement that can reduce dispersion mismatch.

FIG. 13 illustrates an interferometer having a compensating opticalelement that can reduce dispersion mismatch.

FIG. 14 illustrates a compensating optical element that can be used toreduce field dependent dispersion mismatch.

FIG. 15 illustrates a wavelength dependent lateral aberration.

FIG. 16 illustrates a geometry dependent lateral aberration.

FIG. 17 illustrates determining information related to a wavelengthdependent lateral aberration of an optical element.

FIG. 18 illustrates determining information related to a geometrydependent lateral aberration of an optical element.

FIG. 19 illustrates determining information related to a lateralaberration of an interferometer.

FIGS. 20A and 20B illustrate cross sections through phase profiles shownin FIG. 19.

FIG. 21 illustrates a system for determining information related to alateral aberration of an optical element.

FIG. 22 illustrates a system for determining information related to alateral aberration of an optical element.

DETAILED DESCRIPTION

Interferometers are systems that can measure the intensity ofinterference between light (e.g., ultraviolet light, visible light, orinfrared light) reflected from one or more spatial locations of a testobject (e.g., different locations about a surface of the test object)and light reflected from a reference object. The intensity ofinterference depends upon the optical path difference (OPD) between thelight reflected from the test object and the light reflected from thereference object. Typically, an interferometer measures the intensity ofinterference for each of multiple different OPD values (e.g., by movingthe test and/or reference objects through a number of scan position eachcorresponding to a different OPD value). For each spatial location ofthe test object, the interference intensities measured at the differentOPD values define an interference signal. In low coherenceinterferometers, each interference signal includes multiple frequencycomponents, each related to a different wavelength of the interferinglight and/or to a different angle of incidence of the interfering light.Each interference signal can be used to determine information about thetest object (e.g., the height of the spatial location corresponding tothe interference signal). Information from multiple interference signalscan be used to prepare information about the corresponding multiplelocations of the test object (e.g., to prepare an image of the testobject, a phase profile, and/or a height profile of the test object).

In practice, the interferometer can perturb the measured interferenceintensities. One type of perturbation results from differences (e.g.,mismatches) in the thicknesses of optical media (e.g., glass and air)traversed by the light traveling along the test arm of theinterferometer (e.g., the light going to and reflecting from the testobject) from and the light traveling along the reference arm of theinterferometer (e.g., the light going to and reflecting from thereference object). Optical thickness mismatches can result from, forexample, variations in the manufacture of optical elements such aslenses (e.g., lens systems), beam splitters, and wedges that arepositioned along optical paths of the interferometer and/or from theposition of these optical elements (e.g., from their misalignments(e.g., excentrations). The effects tend to be large for interferometersthat have many optical elements not shared by the test and referencepaths. One example is a Linnik interferometer in which the test andreference paths each have a different microscope objective.

Perturbations caused by optical thickness mismatches result from thedependence of the refractive index of the mismatched optical media onthe wavelength of light (e.g., from dispersion). The wavelengthdependence of the refractive index can cause a frequency dependent phaseshift of different frequency components of the interference signals. Ifneglected, the dispersion mismatch can degrade the accuracy andprecision of the information determined about the test object.

Another type of perturbation includes lateral aberrations. Theseaberrations include wavelength dependent aberrations (e.g., lateralcolor) that result from the wavelength dependent focus of opticalelements and the geometry dependent aberrations (e.g., sphericalaberration and coma) that result from geometrical dependent focus (e.g.,angle of incidence dependent focus) of optical elements. Lateralaberrations cause the light reflected from each spatial location of atest object to be focused over a wider area of a detector (e.g., as ablurry image) than in the absence of the aberrations.

As described herein, one or more interference signals can be used todetermine information about one or more optical elements positionedalong one or more optical paths of an interferometer.

In some embodiments, the information relates to dispersion (e.g., to adispersion mismatch). For example, the information may be indicative ofa property (e.g., refractive index and/or thickness) of one or moreoptical elements that can be positioned along an optical path of theinterferometer to compensate for a dispersion mismatch that results fromdifferences in the thickness of optical media traversed by lighttraveling along the test path and light traveling along the referencepath of the interferometer. Based on the information, one or morecompensating optical elements can be positioned along optical paths ofthe interferometer. Once the compensating optical elements having beenpositioned, one or more interference signals can measured. Thesemeasured interference signals can be used to determine whether, forexample, different (e.g., additional or reduced) compensation is needed.For example, an iterative process may be used to determine properties ofa compensating optical element that reduces the optical thicknessmismatch to below a threshold value.

In some embodiments, the information determined from interferencesignals relates to a dispersion (e.g., a dispersion mismatch) thatresults from a position of one or more optics with respect to an opticalaxis of the interferometer. For example, the information may beindicative of a deviation of an optical element from center with respectto an optical axis and/or a deviation in longitudinal spacing betweenoptical elements (e.g., between lenses or lens groups). The position ofthe optical element(s) can be changed and, optionally, interferenceintensities measured with the optical element in the changed position.The measured interference intensities can be used to determine, forexample, whether a different position of the optical element is needed.An iterative process may be used to determine a position of the positionof one or more optical elements that reduces the optical thicknessmismatch to below a threshold value.

In some embodiments, the information relates to a lateral aberration(e.g., to a wavelength dependent lateral aberration and/or to a geometrydependent lateral aberration). The information can be presented as avector map indicative of the lateral aberration over at least some(e.g., most or all) of the field of view of an optical element oroptical system (e.g., of the interferometer). Such information (e.g., avector map) can be prepared by obtaining multiple interference signalsfrom a patterned object (e.g., an object with spatial features such as alateral calibration standard, an object with a random rough surface, oran object with a patterned array). Each of at least two differentfrequency components of the interference signals are used to prepareinformation about the corresponding to multiple locations of the testobject (e.g., to prepare an image of the test object from each frequencycomponent, to prepare a phase profile from each frequency component,and/or to prepare a height profile of the test object from eachfrequency component). Typically, the information prepared using eachfrequency component is indicative of the locations of various surfacefeatures of the test object. However, lateral aberration generallyshifts the locations as determined from the different frequencycomponents. A correlation algorithm (e.g., a cross-correlationalgorithm) can be used to compare the relative locations of surfacefeatures, and, accordingly, determine the extent of lateral aberration.

In some embodiments, the information relates to a modulation transferfunction (MTF) and/or instrument transfer function (ITF) of an opticalelement and/or optical system (e.g., of an interferometer). The MTF andITF are related to the point spread function of an optical system, whichis also referred to as the blur spot resulting from optical andchromatic aberrations. Typically, phase and/or magnitude data determinedfrom different frequency components of multiple interference signals areused to determine information about a patterned object. The MTF and/orITF can be determined based on analysis of height and/or reflectivitydiscontinuities of the patterned object for each of multiple positionswithin the field of view of the optical element or optical system.

Referring now to FIG. 1, an exemplary interferometer system 50 forobtaining interference signals includes an interferometer 51 and aprocessor 52 (e.g., an automated computer control system). Themeasurement system 50 is operable to obtain an interference signal fromeach of multiple spatial locations of a test object 53.

Measurement system 50 includes a light source 54, a first focusing optic(e.g., one or more lenses) 56, a beam splitting element 57, a secondfocusing optic 62, a reference object 58, a third focusing optic 60, anda detector 59. Light source emits 54 emits spectrally-broadband light(e.g., white light), which illuminates a diffusing screen 55. Firstfocusing optic 56 collects light from screen 55 and transmits collimatedlight to beam-splitting element 57, which splits the collimated lightinto first and second portions. A first portion of the collimated lightis received by second focusing optic 62, which focuses the first portionof the light onto reference object 58. Light reflected from thereference object is received by second focusing optic 62, whichtransmits collimated light reflected by the reference object 58 back tobeam-splitting element 57. Beam-splitting element 57 directs the secondportion of the collimated light to third focusing optic 60, whichfocuses the light onto test object 53. Light reflected from test object53 is received by third focusing optic 60, which transmits collimatedlight reflected by test object 53 back to beam-splitting element 57.Beam-splitting element 57 combines light reflected from reference object58 and test object 53 and directs the combined light to a fourthfocusing optic 61, which focuses the combined light to a detector 59.

Detector 59 is typically a multidimensional detector (e.g., a chargecoupled device (CCD) or charge injection device (CID)) having aplurality of detector elements (e.g., pixels) arranged in one or moredimensions (e.g., two dimensions). Optics 60 and 61 focus lightreflected from test object 53 onto detector 59 so that each detectorelement of detector 59 receives light reflected from a correspondingspatial location (e.g., a point or other small region) of test object53. Light reflected from respective spatial locations of test object 53and light reflected from reference object 58 interferes at detector 59.Each detector element produces a detector signal related to theintensity of the interfering light.

System 50 is configured to measure interference signals related tospatial locations of test object 53. Typically, system 50 creates an OPDbetween light reflected from reference object 58 and light reflectedfrom test object 53. For example, test object 53 can be displacedthrough a number of scan positions along a scan dimension axis by a scanmechanism (e.g., an electromechanical transducer 63 (e.g., apiezoelectric transducer (PZT)), and associated drive electronics 64)controlled by computer 52. In some embodiments, a scan positionincrement between successive scan positions is at least about λ/15(e.g., at least about λ/12, at least about λ/10), where λ is a meanwavelength of the light detected at each pixel.

For each scan position, detector 59 outputs an intensity value (e.g.,the intensity detected by a given detector element) for each of multipledifferent spatial locations of the test object. Taken along the scandimension, the intensity values for each spatial location define aninterference signal corresponding to the spatial location. The intensityvalues corresponding to a common scan position define a data set (e.g.,an interferogram) for that scan position. The spectral distribution ofthe light source (e.g., the range of emission wavelengths), thegeometric properties of optical elements of the interferometer (e.g.,the angles of incidence with which the light is received and transmittedby the optical elements), the optical properties of optical elements ofthe interferometer (e.g., the wavelength dependence of the refractiveindex of refractive optical elements), and the spectral response of thedetector define an effective frequency spectrum for interference signalsobtained using the interferometer. This effective spectrum is nominallycentered at a wavenumber k₀.

Referring to FIG. 2, an exemplary interference signal 75 includes aninterference intensity value 77 j for each of N scan positions 79 _(i)along a scan dimension axis 81. The intensity values 77 j ofinterference signal 75 correspond to a single spatial location of a testobject. Intensity values 77 j map out a number of oscillations (e.g.,fringes), which decay on either side of a maximum according to a lowcoherence envelope 83, which does not expressly appear in suchinterference signals but is shown for clarity of discussion. The widthof coherence envelope 83 corresponds generally to the coherence lengthof the detected light, which is related to the effective spatialfrequency spectrum of the interferometer. Among the factors thatdetermine the coherence length are temporal coherence phenomena relatedto, for example, the spectral bandwidth of the light, and spatialcoherence phenomena related to, for example, the range of angles ofincidence of light illuminating the test object. Techniques fordetermining information about a test object based on interferencesignals include transform-based methods (e.g., frequency domain analysis(FDA) as described in U.S. Pat. No. 5,398,113 to de Groot, the contentsof which patent is incorporated herein by reference).

As can be seen from FIG. 2, interference signal 75 results fromdetecting intensity values over a range of scan positions 79 j that isgreater than about ¾ of the width of the coherence envelope. In someembodiments, the intensity values are detected over a range of scanpositions that is greater than the width of the coherence envelope and,therefore, greater than the coherence length of the detected light.

Because the spectral bandwidth of the interfering light includesmultiple wavelengths λ, interference signal 75 includes contributionsfrom interference at multiple frequencies k. Each interference frequencycan be expressed as a wavenumber k=2π/λ, where λ is the wavelength ofthe light that results in interference at the wavenumber k. As discussedabove, a possible interference signal distortion results from acombination of optical thickness mismatches and the wavelengthdependence of the refractive index of optical media traversed by lightreflected from the test object and light reflected from the referenceobject. For each frequency k, a phase shift φ(k) can be expressed as:

$\begin{matrix}{{\phi (k)} = {2\; k{\sum\limits_{i}{{n_{i}(k)}\Delta \; t_{i}}}}} & (1)\end{matrix}$

where n_(i)(k) is the refractive index of the i^(th) medium traversed bylight traveling along the test and reference paths at the wavelengthcorresponding to wavenumber k, and Δt_(i) is the thickness difference ofthe i^(th) medium between the test and reference path.

In an ideal interferometer, the Δt_(i) factors corresponding to opticalmedia (e.g., glass (e.g., BK7, SFL6, fused silica)) are all equal tozero so that no phase shift results from thickness mismatches. Inpractice, the thicknesses of optical media along the test and referencearms are typically at least partially mismatched so that the Δt_(i)factors are non-zero (e.g., the light traveling along one of the testand reference arms travels a greater distance through at least oneoptical medium than the light traveling along the other arm). Asdiscussed next, however, an interference signal can be used to determineproperties (e.g., refractive index and thickness) of a compensatingoptical element that, when positioned along an optical path of theinterferometer, reduces the n_(i)(k)Δt_(i) factors (e.g., causes then_(i)(k)Δt_(i) factors to sum to about zero) and can reduce (e.g.,eliminate) optical thickness mismatches of an interferometer.

Expressing Eq. as a Taylor expansion about the central source frequencyk₀ yields:

$\begin{matrix}{{\phi (k)} \approx {{2\; k_{0}{\sum\limits_{i}{{n_{i}\left( k_{0} \right)}\Delta \; t_{i}}}} + {2\left( {k - k_{0}} \right){\sum\limits_{i}\left( {{{n_{i}\left( k_{0} \right)}\Delta \; t_{i}} + {k_{0}\frac{\partial n_{i}}{\partial k}\left( k_{0} \right)\Delta \; t_{i}}} \right)}} + {\left( {k - k_{0}} \right)^{2}{\sum\limits_{i}\left( {{2\frac{\partial n_{i}}{\partial k}\left( k_{0} \right)\Delta \; t_{i}} + {k_{0}\frac{\partial^{2}n_{i}}{\partial k^{2}}\left( k_{0} \right)\Delta \; t_{i}}} \right)}} + \ldots}} & (2)\end{matrix}$

where the term ∂ni/∂k is the 1^(st) derivative of the refractive indexof the ith optical medium with respect to the wavelength correspondingto the interference frequency k and the term ∂²ni/∂k² is the 2^(nd)derivative of the refractive index of the ith optical medium withrespect to the wavelength corresponding to the interference frequency k.The first term in the Taylor expansion is a constant phase factor. Thesecond term is a linear function of wavenumber k, which is related tothe group velocity index and corresponds to a shift of the position atwhich maximum fringe contrast of the interference signal is observed asa function of interferometer scan position. The second term does notaffect interference contrast or the shape of the modulation envelope.The third term in the Taylor expansion, however, represents nonlinearity(e.g., parabolic nonlinearity) of the relationship between phase andfrequency components of the interference signal. Such nonlinearity isindicative of a dispersion mismatch of the test and reference arms ofthe interferometer. The phase shifts, which are typically different foreach frequency k, reduce the contrast of the interference signal andspread the interference signal over a larger range of OPD than wouldotherwise be observed. As discussed next, the third and/or higher termsof Eq. 3 can be used to determine information about the opticalthickness mismatches and to determine, for example, properties (e.g.,1st and 2nd order derivatives of refractive index with respect towavelength, thickness. and/or shape) of compensating optical elementsthat reduce (e.g., eliminate) the phase shifts caused by the mismatches.

Typically, the method for determining information about opticalthickness mismatches includes obtaining one or more interference signalsusing an interferometer. Each interference signal corresponds to adifferent spatial location of a test object. For each interferencesignal, the phase of each of multiple frequencies of the interferencesignal is determined. For example, the phase of different frequencies ofan interference signal can be determined by Fourier transformation ofthe interference signal. The phases of different frequencies of theinterference signal are fit (e.g., by least squares) to function thatincludes one or more fitting parameters that are related to the opticalthickness mismatches between the test and reference arms of theinterferometer. The fitted parameters are used to determine a property(e.g., 1st and 2nd order derivatives of refractive index with respect towavelength, thickness, and/or shape) of one or more optical elements(e.g., plates, wedges, lenses, beam splitters) that can be positionedalong an optical path of the interferometer to compensate for (e.g.,reduce or eliminate) the dispersion mismatch.

We now discuss an exemplary method for determining information aboutoptical thickness mismatch beginning by presenting an interferencesignal obtained with an interferometer exhibiting such mismatches.

Referring to FIG. 3, an interference signal 100 is an example of aninterference signal that corresponds to a single spatial location of anobject as obtained using an interferometer having an endoscopicobjective positioned along each arm. The interferometer uses a broadbandlight source. Such interferometers are typically used to measure spheresand cones, and are similar to Linnik interferometers in that manyoptical elements are positioned along both the test and reference armsof the interferometer. Interference signal 100 includes intensity valuesmeasured over a range of scan positions corresponding to an OPD of about30 microns, but exhibits substantial interference intensity over asmaller range r1 corresponding to an OPD of about 20 microns.

Interference signal 100 has a fringe contrast C given by:

$\begin{matrix}{C = \frac{{\max (s)} - {\min (s)}}{{\max (s)} + {\min (s)}}} & (3)\end{matrix}$

where s is an intensity of the measured interference signal. Thecontrast C, as defined by equation (3) is equal to 23.8%.

Continuing with the method, interference signal 100 is transformed(e.g., by Fourier transformation) into an inverse dimension with respectto the values for the OPD's along the scan dimension. The phase isdetermined for each of multiple frequencies of the transformedinterference signal. Typically, most (e.g., all) of the multiplefrequencies for which the phase is determined correspond to frequenciesthat correspond to different wavelengths of the spectrum of the lightsource of the interferometer.

FIG. 4 shows a plot of phase φ(k) vs. frequency data (in units of cyclesequal to 2π radians) for each of N different frequencies k of theinterference signal. The plot shows a clear nonlinear (e.g., parabolic)shape due to an optical medium thickness imbalance in the two arms ofthe interferometer used to obtain interference signal 100. The phasesφ(k) in FIG. 4 are fit to a second order polynomial function accordingto:

φ(k)=A ₀ +A ₁(k−k ₀)+A ₂(k−k ₀)²  4

where k₀ is the central wavelength of the effective spatial frequencyspectrum of the interferometer and A₀, A₁, and A₂ are the fittingparameters. Such a fit can be performed by, for example, a weightedleast squares fitting routine with each phase φ(k) weighted by thesquare of the Fourier amplitude of the corresponding frequency k. Thesolid line in FIG. 4 is determined from the fitting parametersdetermined from a weighted least squares fit of Eq. 4 to the phasesφ(k).

The fitting parameters A₀, A₁, and A₂ correspond to the first, second,and third order components of the Taylor expansion seen in Eq. 2. Inparticular, the third order fitting parameter A₂ can be expressed as:

$\begin{matrix}{A_{2} \approx {\sum\limits_{i}\left( {{2\frac{\partial n_{i}}{\partial k}\left( k_{0} \right)\Delta \; t_{i}} + {k_{0}\frac{\partial^{2}n_{i}}{\partial k^{2}}\left( k_{0} \right)\Delta \; t_{i}}} \right)}} & 5\end{matrix}$

Eq. 5 can be rearranged to determine a thickness Δtc of a single opticalmedium (e.g., an optical glass (e.g., BK7) or fused silica (e.g., SF6))that compensates for the optical thickness mismatch between the test andreference arms of the interferometer:

$\begin{matrix}{{\Delta \; t_{c}} = \frac{- A_{2}}{{2\frac{\partial n_{c}}{\partial k}\left( k_{0} \right)} + {k_{0}\frac{\partial^{2}n_{c}}{\partial k^{2}}\left( k_{0} \right)}}} & 6\end{matrix}$

where ∂nc/∂k is the 1^(st) derivative of the refractive index of thecompensating medium with respect to the wavelength corresponding tofrequency k and the term ∂²nc/∂k² is the 2^(nd) derivative of therefractive index of the compensating medium with respect to thewavelength corresponding to frequency k. For a given optical medium, the∂nc/∂k and ∂²nc/∂k² can be determined from its Sellmeier coefficients,which describe the relationship between refractive index and wavelengthfor an optical medium.

To use Eq. 6, a compensating optical medium (e.g., an optical glass(e.g., BK7) or fused silica (e.g., SFL6)) is typically selected. The∂nc/∂k and ∂²nc/∂k² of the optical medium are determined (e.g., from theSellmeier coefficients of the medium) and substituted into Eq. 6, whichis then solved for the thickness Δtc. Based on the fit of Eq. 4 to thephases φ(k) of FIG. 4, Eq. 6 predicts that a BK7 optical element (e.g.,optical plate) having a thickness Δtc of about 190 microns willcompensate for the optical thickness mismatch of the interferometer usedto obtain interference signal 100 if the optical element is positionedalong one of the arms of the interferometer. To compensate for theoptical thickness mismatch, the compensating optical element istypically positioned along the interferometer arm in which the totaloptical medium thickness is too small.

The interferometer arm along which the compensating optical elementshould be positioned to compensate for the thickness mismatch can beidentified by taking into account the arm along which the OPD is varied(e.g., scanned) to obtain the interference signal, the sign of the OPDvariation, and the sign convention used for determining the inversetransform of the interference signal. Alternatively, a plot of φ(k) vs.frequency can be determined from an interference signal obtained withthe compensating optic positioned along one of the interferometer arms.The non-linearity of the φ(k) data shows readily if the correct arm iscompensated.

The extent of nonlinearity of phase φ(k) vs. frequency data can beexpressed as an objective parameter εφ. For example, εφ can determinedbased on the difference between the phases φ(k) and a fit to the phases(e.g., the difference between the phases φ(k) and the least squares fitof Eq. 4 to the phases). In some embodiments, εφ is given by thepeak-to-valley range of the difference between phases φ(k) and a fit tothe phases. In some embodiments, εφ is related to the squareddifferences between the phases φ(k) and a fit to the phases. Forexample, εφ can be expressed as the χ² sum of residuals or the standarddeviation between the phases φ(k) and a fit to the phases. Typically,the optical thickness mismatch compensation method aims at reducing εφbelow a given threshold.

FIG. 5 shows an interference signal 105 obtained with the sameinterferometer used to obtain interference signal 100 of FIG. 3 but witha 210-micron thick BK7 plate positioned along the interferometer arm inwhich the total optical medium thickness was too small. Interferencesignal 105 has a higher contrast C of 33.3% than interference signal 100(FIG. 3) and exhibits substantial interference intensity over a range r2of OPD variation that is narrower than the range r1 of interferencesignal 100. In low coherence interferometry, a narrower higher contrastinterference signal (e.g., interference signal 105) typically providesbetter precision and accuracy than a wider lower contrast interferencesignal (e.g., interference signal 100).

FIG. 6 shows a phase φ(k) vs. frequency plot as determined from theFourier transform of interference signal 105 (FIG. 5). The phase φ(k)vs. frequency data of FIG. 6 exhibits substantially less non-linearity(e.g., less curvature) than the phase φ(k) vs. frequency data (FIG. 4)determined from interference signal 100 (FIG. 3).

Because interference signal 105 was obtained with a compensating optichaving a thickness of 210 microns rather than the optimal 190 micronspredicted by Eq. 6, the phases φ(k) of FIG. 6 exhibit some residualnonlinearity (e.g., negative curvature) with respect to frequency. Basedon a least squares fit of Eq. 4 to the phases φ(k) determined frominterference signal 105 (this fit is shown as the solid line of FIG. 6),Eq. 6 predicts that a BK7 optical element (e.g., optical plate) having athickness Δtc=20 microns (e.g., 210 microns-190 microns) can bepositioned along the other arm of the interferometer to compensate forthe optical thickness mismatch introduced by the too thick compensatingoptical element. Of course, one could alternatively replace the 210microns thick compensating optical plate with a plate 190 microns thick.

As discussed above, the effective spectrum of an interferometer candepend on both temporal coherence effects related to the source spectralwidth and spatial coherence effects related to the source dimensions andangles of incidence. Information related to the optical thicknessmismatch of such interferometers is typically determined by reducingspatial coherence effects (e.g., by using a sufficiently small sourceemitting area). Once the optical thickness mismatch has been determinedand compensated, the source dimension is restored to whatever size isrequired for normal operation of the interferometer.

While Eq. 6 has been described as predicting a property (e.g., athickness Δtc) of a single compensating optical element, more than oneoptical element can be used to compensate for optical thicknessmismatch. For example, a different optical element (e.g., plate) can beplaced along each arm of the interferometer, where the difference inthicknesses of the optical elements corresponds to Δtc. As an example,if Δtc is predicted to be 75 microns, an optical element having athickness of 500 microns can be positioned along one arm of theinterferometer and an optical element having a thickness of 575 micronscan be positioned along the other arm of the interferometer.

While Eq. 6 has been described as being applied to a compensatingoptical medium with known optical properties (e.g., known 1^(st) and2^(nd) order derivatives of refractive index with respect towavelength), other methods can be used. For example, Eq. 6 can be usedeven if the optical properties of the optical medium are not known.Typically, phase φ(k) vs. frequency data (e.g., as shown in FIG. 4) aredetermined from one or more interference signals obtained using aninterferometer. Phase φ(k) vs. frequency data are also determined fromone or more interference signals obtained using the interferometer witha compensating optical element positioned along one of the optical armsof the interferometer. Here, the compensating optical element typicallyhas a known thickness and shape, but may have other unknown opticalproperties (e.g., unknown 1^(st) and 2^(nd) order derivatives ofrefractive index with respect to wavelength). Eq. 4 is fit to the φ(k)vs. frequency data determined with and without the compensating opticalelement. The unknown optical properties (e.g., the 1^(st) and 2^(nd)order derivatives of refractive index with respect to wavelength) of theoptical medium are determined from the fitted parameters (e.g., from thefitted parameters A₂). Alternatively, a similar method can be used todetermine a thickness and/or shape of an optical element having knownrefractive index properties.

While the method for compensating for optical thickness mismatch hasbeen described as using the second order term of the Taylor expansion ofEq. 2, other orders can be used. For example, phase φ(k) vs. frequencydata (e.g., FIG. 4) can be fit to a function (e.g., a polynomial ofhigher than 2^(nd) order) having more or different parameters than Eq.4. The fitted parameters are related to higher order terms of the Taylorexpansion of Eq. 2 (e.g., 3^(rd), 4^(th), 5^(th), and/or higher orderterms) just as Eq. 6 relates the fitted parameter A₂ of Eq. 4 to the2^(nd) order term of the Taylor expansion in Eq. 3. The relationshipbetween the fitted parameters and the higher order terms of the Taylorexpansion are used to determine optical properties of compensatingmedia.

While a method for determining a property (e.g., a thickness and/orderivatives of refractive index with respect to wavelength) of acompensating optical element has been described as including a fit tophase φ(k) vs. frequency data, other methods can be used. For example,in some embodiments, properties of a compensating optical element thatreduces (e.g., eliminates) optical thickness mismatch are determinedwithout fitting phase vs. frequency data. The method can be performediteratively. Typically, the method includes providing one or more firstinterference signals obtained using an interferometer. Phase φ(k) vs.frequency data are determined from the one or more first interferencesignals. A compensating optic is introduced along one or both of thearms of the interferometer. One or more second interference signals areobtained with the compensating optic in position. Phase φ(k) vs.frequency data are determined from the one or more second interferencesignals. The next step of the method typically includes comparing thephase φ(k) vs. frequency data obtained from the first and secondinterference signals to determine whether the compensating opticincreased or decreased the optical thickness mismatch. Such adetermination can be made, for example, based on the extent ofnonlinearity of the phase φ(k) vs. frequency data. Typically, decreasednonlinearity of the phase φ(k) vs. frequency data indicates decreasedoptical thickness mismatch. The process of obtaining interferencesignals with different compensating optical elements (e.g., opticalelements of different thickness or material) can continue iterativelyuntil phase φ(k) vs. frequency data having a desired degree of linearityare obtained. Hence, the nonlinearity of the phase φ(k) vs. frequencydata can act as a feedback mechanism for conducting an iterativecompensation of optical thickness mismatch.

While optical thickness mismatch compensation has been described asusing one or more compensating optical elements each formed of the sameoptical medium, more than one optical medium can be used to compensatefor optical thickness. Use of more than one optical medium can bebeneficial with, for example, interferometers having many differentoptical elements located on the test and reference arms. A compensatingoptical element of a single optical medium may not sufficiently reduce(e.g., eliminate) the optical thickness mismatch of suchinterferometers.

Typically, the method for using more than one optical medium tocompensate for optical thickness mismatch includes using aninterferometer to obtain one or more interference signals anddetermining phase φ(k) vs. frequency data from the one or moreinterference signals. The phase φ(k) vs. frequency data are fit to afunction having parameters that correspond to compensating opticalelements of different optical media (e.g., different types of opticalglass or fused silica). Properties (e.g., refractive index, thickness,and/or shape) of the compensating optical elements are determined fromthe fitted parameters. Compensating optical elements of two or moredifferent media are positioned along one or more optical paths of theinterferometer. The interferometer is used to obtain one or moreinterference signals with the compensating optical elements in positionand phase φ(k) vs. frequency data are determined from the interferencesignal(s). The remaining optical thickness mismatch can be evaluated by,for example, the parameter εφ to determine whether differentcompensation is needed.

We now discuss an example for compensating optical thickness mismatchusing different optical media by comparing results obtained fromcompensation with a single optical medium and results obtained bycompensation with two optical media.

Referring to FIG. 7, an interference signal 107 is simulated as ifobtained from a Linnik interferometer having a 100× microscope objective(0.94 NA) positioned along its test arm and along its reference arm.Each simulated objective included eleven optical elements (e.g.,lenses). Small thickness variations were introduced into the elevenelements.

FIG. 8 illustrates phase φ(k) vs. frequency data determined frominterference signal 107 by Fourier transformation into an inversedimension with respect to OPD. Eq. 4 was fit to the phase φ(k) vs.frequency data of FIG. 8 and the thickness Δtc of a compensating opticalelement formed of a single optical medium (SFL6) determined using Eq. 6.

Referring to FIG. 9, an interference signal 109 is simulated as ifobtained from the Linnik interferometer used to obtain interferencesignal 107 of FIG. 7 but with the single compensating optical element inposition. The contrast C of interference signal 109 is 94.3% compared tothe contrast C of interference signal 107 of 35.6%.

Referring to FIG. 10, a line 111 indicates the difference in units ofdegrees between the phase φ(k) vs. frequency data of FIG. 8 and a fitdetermined from parameters A₀, A₁, and A₂, of Eq. 4. The extent ofnonlinearity εφ₁ of line 111 as determined by the peak-to-valley rangeof the difference between phases φ(k) and the fit to the phases is about20°.

The thicknesses of compensating optical elements of two differentoptical media can be determined by fitting phase φ(k) vs. frequency datato function that includes the refractive index as a function offrequency k for each optical medium:

φ(k)=B ₀ +B ₁ k+M ₁(k)Δt ₁ +M ₂(k)Δt ₂  7

where B₀ and B₁ are constants, Δt₁ is the thickness of the compensatingoptical element formed of the first optical medium, Δt₂ is the thicknessof the compensating optical element formed of the second optical medium,and M₁(k) and M₂(k) are related to the index of refraction of eachoptical medium by:

M _(i)(k)=2k(n _(i)(k)−n _(ambient)(k) =1,2  8

where n_(ambient) is the index of refraction of the medium where thecorrection takes place (e.g., air) and n_(i) is the refractive index ofthe compensating optical medium at the wavelength of light correspondingto frequency k.

Eq. 7 can be fit to the phase φ(k) vs. frequency data by, for example, aleast squares fitting routine that minimizes the sum:

$\begin{matrix}{\chi^{2} = {\sum\limits_{k}\left( {B_{0} + {B_{1}k} + {{M_{1}(k)}\Delta \; t_{1}} + {{M_{2}(k)}\Delta \; t_{2}} - {\phi (k)}} \right)^{2}}} & 9\end{matrix}$

Eq. 7 is fit to the phase φ(k) vs. frequency data of FIG. 8 using BK7 inaddition to SFL6. As seen in FIG. 10, a line 113 indicates thedifference in units of degrees between the phase φ(k) vs. frequency dataof FIG. 8 and the fit determined from parameters of Eq. 7. The extent ofnonlinearity εφ₂ of line 113 as determined by the peak-to-valley rangeof the difference between phases φ(k) and the fit to the phases is about5°.

We now discuss various methods for positioning one or more compensatingoptical elements along an optical path of an interferometer tocompensate for optical thickness mismatch.

Referring to FIG. 11, an example of compensating for optical thicknessmismatch is shown for an interferometer 250 having a beam splitterformed as a plate 204′, a test arm 203′, and a reference arm 205′. Acompensating optical element (e.g., a plate 216) is positioned alongtest arm 203′. Plate 216 can be rotated about an axis perpendicular toan optical axis 218 of interferometer 250 to introduce a variablecompensation for optical thickness mismatch.

Referring to FIG. 12, a Michelson interferometer is an example of aninterferometer that can be used to obtain information about a testobject 202. Light from a light source 201 is collimated and distributedby a beam splitter 204 between a test arm 203 and reference arm 205 ofthe interferometer. The light reflects from the light reflects from thetest object 202 and a reference object 207, light is recombined by beamsplitter 204 and interferes at a detector 209.

One source of optical thickness mismatch is beam splitter 204 becauselight traveling along the test arm 203 typically traverses a differentthickness of the beam splitter than light traveling along the referencearm 205. One method for correcting the optical thickness mismatch is tosplit the beam splitter into first and second halves 211, 213. Each half211, 213 can be slid with respect to the other to modify the thicknessof the beam splitter traversed by light traveling along the test andreference arms.

Interferometers having beam splitters are described in U.S. patentapplication Ser. No. 10/659,060 filed Sep. 9, 2003 by Peter de Groot,which application is incorporated herein by reference.

As an alternative or in combination to beam splitter 204, an adjustableoptical element (e.g., a split plate 215) can be positioned along one ofthe test and reference arms. Plate 215 includes first and second wedges217, 219 that can be translated with respect to one another to modifythe thickness traversed by the light traveling along reference arm 205.Typically, a fixed plate of the same nominal thickness as the splitplate is positioned other arm.

If more than one compensating optical medium is used, the methodsdescribed above can be combined. For example, a split beam splitter orplate formed of two halves of a different optical medium can be used.

In another example, an optical plate is positioned along each arm of aninterferometer. Each plate has the same nominal thickness. A correctionis introduced by a rotation of one plate about an axis perpendicular tothe optical axis of the system.

Referring to FIG. 13, a Linnik interferometer 275 includes first andsecond objectives 277, 279 respectively positioned along test andreference arms 281, 283 of the interferometer. Each objective includes aplurality of optical elements (e.g., multiple lenses). Typically,compensating optical elements (e.g., beam splitter 204, split plate 215,and/or rotating plate 285) are positioned at locations in which lightrays that originate from individual object points are parallel to oneanother.

While configurations for correcting optical thickness mismatch have beendescribed as being uniform across a field of view of an interferometer,other configurations can be used. For example, in some embodiments, theoptical thickness mismatch determined from interference signals thatcorrespond to different spatial locations of the test object may bedifferent (e.g., the optical thickness mismatch may vary across thefield of view of the interferometer). The presence of a field dependentoptical thickness mismatch can be determined based on multipleinterference signals each corresponding to a different spatial locationof the test object. As discussed next, a field dependent opticalthickness mismatch can be compensated for by using one or more opticalelements having a property (e.g., a shape, a thickness, and/orrefractive index) that varies as a function of position in the field ofview.

As seen in FIG. 14, a compensating optical element 300 includes firstand second prisms 301, 303 that can be translated with respect to oneanother (e.g., along an axis 305) and rotated with respect to oneanother about an optical axis of an interferometer (e.g., about an axis307). Additionally, optical element 300 can be rotated in its entiretyabout the optical axis of the interferometer.

The nominal thickness of optical element 300 can be changed bytranslating of one of prisms 301, 303 with respect to the other prism toadjust the average optical thickness mismatch compensation across thefield the view. The field position dependent thickness of opticalelement 300 can be changed by rotating one of prisms 301, 303 about axis307 with respect to the other prism to introduce a field dependentoptical thickness mismatch compensation. Typically, an optical elementfor providing field dependent optical thickness mismatch compensation ispositioned at an intermediate image of the object or is positioned closeto the test object.

While methods for compensating for optical thickness mismatch have beendescribed as including the use of a compensating optical element, othermethods can be used. For example, in some embodiments, the informationabout an optical thickness mismatch is related to the position of anoptical element (e.g., a decenter, a tip, a tilt, and/or a longitudinalspacing of the optical element). Typically, the optical element is arefractive optical element (e.g., a lens or lens system (e.g., anobjective). Because refractive elements can introduce varying amounts ofoptical thickness as a function of the position of the refractiveelement with respect to the optical axis of the interferometer, it canbe important to match this field-dependent parameter in the test andreference arms in order to obtain high-contrast interference signals.The performance of interferometers such as Linnik or Twyman-Greeninterferometers that include multiple refractive elements isparticularly sensitive to the position of optical elements.

Typically, a method for determining information about an opticalthickness mismatch related to a position of an optical element includesobtaining interference signals from each of multiple spatial locationsof a test object using an interferometer. Phase φ(k) vs. frequency dataare determined for each of the interference signals. The extent ofnon-linearity (e.g., εφ) is determined for each phase φ(k) vs. frequencydata and mapped according to position within the field of theinterferometer. For a perfectly aligned interferometer, the map ofextent of the non-linearity of the phase φ(k) vs. frequency data isrotationally symmetrical function about the optical axis of theinterferometer. If a refractive component (e.g., an lens or lens system(e.g., an objective) is misaligned in one of the arms of theinterferometer (e.g., decentered, tipped, and/or tilted), then thenon-linearity phase map will no longer be symmetrical. For example,decenter of a component yields to first order a tilt of thenon-linearity map. This overall effect can be understood from the factthat optical thickness mismatches can typically be modeled as aspherical function. Decenter of one objective results in a shear of thisspherical function, resulting in a tilted plane. In those cases, thegradient of the non-linearity map can be used as a quantitative feedbacksignal for alignment. The optical element can be repositioned based onthe information about the optical thickness mismatch and an iterativemethod used to determine the optimal position.

In some situations, the map of the phase nonlinearity as a function offield position includes more complicated radial patterns such that theeffect of a decenter is no longer well approximated by a tilt.Nevertheless, the argument that the map should be a symmetrical functionwhen alignment is optimum still holds. Hence, more advanced processingof the map of the phase nonlinearity as a function of field positionusing, for example, Zemike polynomials, or simple visual observation,can still provide quantitative or qualitative information aboutmisalignment of one or more optical elements.

As discussed above, lateral aberrations are another source of error thatcan be introduced by interferometers. We next discuss examples oflateral aberrations and methods for determining the presence of lateralaberrations and reducing the same.

Referring back to FIG. 1, detector 59 of interferometer system 50includes multiple detector elements. In an ideal interferometer, theinterferometer focuses light reflected from different spatial locationsof the test object to sharp, distinct foci at different detectorelements. In practice, the foci corresponding to different spatiallocations of the test object can contain some amount of aberration(e.g., blur). Lateral aberration is one type of aberration that can blurthe foci corresponding to spatial locations of a test object.

FIG. 15 illustrates lateral color, a wavelength dependent lateralaberration. Lateral color results because refractive optics (e.g.,lenses and lens systems) typically focus light of different wavelengthsat different lateral locations with respect to the optical axis. Forexample, FIG. 15 shows a light beam 326 having first and seconddifferent wavelengths reflected from a spatial location 327 of a testobject 325. A lens 329 focuses light beam 326 to a focus 331, whichincludes a spot 333 corresponding to light of the first wavelength and aspot 335 corresponding to light of the second wavelength of beam 326.Spots 333, 335 are spaced apart by a distance 337 that corresponds to awavelength dependent lateral aberration of lens 329. In aninterferometer, light reflected from each spatial location of a testobject may be focused into a blurry focus including multiple spots eachcorresponding to different wavelength components of the illuminatinglight.

FIG. 16 illustrates coma, a geometry dependent lateral aberration. Comaresults because optics (e.g., lenses and lens systems) focus light to alocation and shape that depends on the geometry of the path traveled bythe focused light. For example, FIG. 16 shows a monochromatic light beam338 reflected from spatial location 327 of test object 325. Forillustration purposes, a spatial filter having first and second annuli345, 347 is positioned between object 325 and lens 329. Annulus 345allows only light having small angles of incidence to pass through.Annulus 347 allows only light have a large angle of incidence to passthrough. Lens 329 focuses light passing through first and second annuli345, 347 to a focus 341 including a spot 343 corresponding to lightpassing through annulus 345 and a spot 355 corresponding to lightpassing through annulus 347. Spots 343, 345 are spaced apart by adistance 351 that corresponds to a geometry dependent lateral aberrationof lens 329. In an interferometer, light reflected from each spatiallocation of a test object may be focused into a blurry focus includingmultiple spots each corresponding to different angles of incidencetraversed by the illuminating and/or reflected light.

As discussed above, low coherence interference signals typically includeinterference at each of multiple frequencies k that correspond to thespectral distribution of the light source (e.g., the range of emissionwavelengths) and the geometric properties of optical elements of theinterferometer (e.g., the angles of incidence with which the light isreceived and transmitted by the optical elements of the interferometer).One limiting case is an interferometer that uses a low numericalaperture illumination and imaging system, and a relatively broad sourcespectrum. In such interferometers, the range of illumination wavelengthsmay be large and the range of angles of incidence is small. Thefrequencies k of interference signals obtained with theseinterferometers correspond to the spectral distribution of the lightsource. Consequently, the lateral aberrations in such interferometerstend to be dominated by wavelength dependent lateral aberrations (e.g.,as shown in FIG. 15).

Another limiting case is a high-numerical aperture interferometer wherethe light source is spatially extended but is nominally monochromatic.In such interferometers, the range of wavelengths is small and the rangeof angles of incidence is large. The frequencies k of interferencesignals obtained with these interferometers correspond to the spatialfrequencies resulting from geometric properties of optical elements ofthe interferometer. Consequently, the lateral aberrations in suchinterferometers tend to be dominated by geometry dependent lateralaberrations (e.g., as shown in FIG. 16).

We next discuss methods for determining information related to lateralaberration of one or more optical elements (e.g., a lens or lenssystem). The optical element(s) can be, for example, an optical elementof an interferometer or an optical element to be tested (e.g., duringmanufacture). Typically, the methods include using the optical elementto determine information related to multiple spatial locations of apatterned test object (e.g., determining an image, phase profile, and/orheight profile of the test object) for each of first and secondillumination conditions (e.g., different illumination wavelengths ordifferent illumination geometries (e.g., different angles ofincidence)). Information related to lateral aberrations is determinedfrom the information related to the multiple spatial locationsdetermined under the different illumination conditions. In general, thedetermination includes a cross-correlation of information determinedunder the first illumination condition and information determined underthe second illumination condition.

With reference to FIG. 17, some embodiments of the method includedetermining information related to wavelength dependent lateralaberrations of an optical element (e.g., a lens system 400). Typically,the optical element is configured to image a patterned test object 400onto a detector 403 having multiple detector elements (e.g., a CCDhaving multiple pixels). Test object 401 includes multiple surfacefeatures 402 i (e.g., multiple step features). A spectral filter 405 isused to illuminate the object with generally monochromatic light havinga first central wavelength. Typically, the test object is illuminatedwith a narrow range of angles of incidence. An image 407 of the testobject is obtained with the light having the first central wavelength.Spectral filter 405 is then configured to illuminate the test objectwith generally monochromatic light having a different centralwavelength. An image 409 of the test object is obtained with the lighthaving the second central wavelength.

Image 407 includes image features 408 i corresponding to surfacefeatures 402 i of patterned test object 401. Image 409 includes imagefeatures 410 i corresponding to surface features 402 i of patterned testobject 401. Because of wavelength dependent lateral aberrations, imagefeatures 408 i and image features 410 i are shifted with respect to oneanother. For example, a vector 411 indicates the magnitude and directionof the image feature shift between image feature 408 i of image 407(corresponding to surface feature 402 i) and image feature 410 i ofimage 409 (also corresponding to surface feature 402 i). Vector 411 isindicative of a wavelength dependent lateral aberration of opticalelement 400 at a position within its field of view corresponding tosurface feature 402 i.

The wavelength dependent lateral aberration at a field positioncorresponding to surface feature 402 i (e.g., vector 411) can bedetermined by cross-correlating a sub-region 412 i of image 407 with asub-region 413 i of image 409. Sub-regions 412 i and 413 i are nominallycentered on the same surface feature (e.g., surface feature 402 i)) andhave the same width and height in terms of detector pixels. Thecross-correlation typically includes transforming each image sub-regioninto an inverse domain. The transformation is generally atwo-dimensional transformation and can be accomplished by, for example,Fourier transformation. Vector 411 is determined based on thetransformed sub-regions.

In some embodiments, the cross-correlation includes determining a phasemap corresponding to each transformed sub-region. Phase unwrapping isused to prepare an unwrapped phase map that is a linear function ofspatial frequency. The phase gradient in the vertical and horizontaldirections yields the magnitude and direction of the shift (e.g., vector411) of the surface feature between the transformed sub-regions.

In some embodiments, the cross-correlation includes determining theproduct of the Fourier transform of one sub-region and the conjugate ofthe Fourier transform of the other sub-region. The product is inversetransformed (e.g., by inverse Fourier transformation). The magnitude anddirection of the shift of the surface features (e.g., vector 411) can bedetermined from a peak in the inverse transformed product.

In general, the image sub-regions are scaled prior to transformation tomake them as similar as possible in amplitude. An image sub-region canbe scaled by, for example, subtracting the average of the sub-regionfrom each pixel of that sub-region and then dividing each pixel by thestandard deviation of the average subtracted pixels.

The wavelength dependent lateral aberration of optical element 400 canbe determined for each of multiple field positions by determining themagnitude and direction of the shift corresponding to image sub-regionsdistributed across the field of view of the optical element. Asdiscussed below, the lateral aberrations so determined can be expressedas a vector map indicative of the field-dependent lateral aberration.

With reference to FIG. 18, some embodiments of the method includedetermining information related to geometry dependent lateralaberrations of an optical element (e.g., of lens system 400). Theoptical element is configured to image patterned test object 400 ontodetector 403. Typically, the test object is illuminated by generallymonochromatic light having a wide range of angles of incidence. An image407′ of the test object is obtained using a first annulus 421 thataccepts only light having a narrow range of angles of incidence centeredabout a first angle θ. An image 409′ of the test object is obtainedusing a second annulus 423 that accepts only light having a narrow rangeof angles of incidence centered about a second angle different fromangle θ.

Image 407′ includes image features 408′i corresponding to surfacefeatures 402 i of patterned test object 401. Image 409′ includes imagefeatures 410′i corresponding to surface features 402 i of patterned testobject 401. Because of geometry dependent lateral aberrations, imagefeatures 408′i and image features 410′i are shifted with respect to oneanother. For example, a vector 411′ indicates the magnitude anddirection of the image feature shift between image feature 408′i ofimage 407′ (corresponding to surface feature 402 i) and image feature410′i of image 409′ (also corresponding to surface feature 402 i).Vector 411′ is indicative of a geometry dependent lateral aberration ofoptical element 400 at a position within its field of view correspondingto surface feature 402 i.

Vector 411′ can be determined using correlation techniques as describedabove. Information related to geometry dependent lateral aberrations ofoptical element 400 can be determined for multiple positions within thefield of view of the optical element (e.g., a vector map of the geometrydependent lateral aberration can be prepared).

With reference to FIG. 19, some embodiments of the method include usingan interferometer to determine information related to a lateralaberration of an optical element (e.g., an optical element(s) of theinterferometer or a test optical element(s)). The method typicallyincludes obtaining an interference signal from each of multiple spatiallocations of a patterned test object 377. Typically, the interferometeris configured so that either wavelength dependent lateral aberration(e.g., a generally low numerical aperture and generally broad sourcespectrum) or geometry dependent lateral aberration (e.g., a generallyhigh numerical aperture and generally narrow source spectrum) willdominate. By way of example, FIG. 19 shows an interference signal 375corresponding to a single spatial location 379 of the test object 377obtained under conditions in which wavelength dependent lateralaberration will dominate.

Each interference signal is transformed into an inverse dimension (e.g.,by Fourier transformation) with respect to OPD to prepare a transformedinterference signal. By way of example, FIG. 19 shows a transformedinterference signal 381 corresponding to the transform of interferencesignal 375.

Transformed interference signal 381 includes a phase component 383 andan amplitude component 385. The phase component 383 is indicative of thephase φ(k) of each frequency k of the interference signal. The amplitudecomponent 385 is indicative of the amplitude of each frequency k of theinterference signal. For each transformed interference signal, the phaseφ(k) and/or amplitude components at each of two different frequencies kare used to determine information related to multiple spatial locationsof patterned test object 377 (e.g., a phase profile, and/or heightprofile of the test object). For example, a phase profile 383 isprepared based on the phase φ(k) and/or amplitude at a frequency k1corresponding to a source wavelength of 516 nanometers and a phaseprofile 385 is prepared based on the phase φ(k) and/or amplitude at afrequency k2 corresponding to a source wavelength 618 of nanometers.FIG. 20A illustrates a cross section 383 a through sub region 383(dashed line) and a cross section 385 a through sub region 385 (solidline) each along the y-dimension. FIG. 20B illustrates a cross section383 b through sub region 383 (dashed line) and a cross section 385 bthrough sub region 385 (solid line) each along the x-dimension.

Pairs of corresponding sub-regions of each phase profile are selected(e.g., based on a regular arrangement of sub-regions across the phaseprofiles). Each pair of corresponding sub-regions includes informationrelated to the same surface feature and is centered about the samenominal location of the test object. For example, a sub-region 384includes phase profile information related to a surface feature 386 of377 as determined from frequency k1 and a sub-region 386 includes phaseprofile information related to a surface feature 386 of 377 asdetermined from frequency k2. Corresponding sub-regions of phaseprofiles 383 and 385 are cross-correlated to determine informationrelated to the lateral aberrations of the interferometer. For example, avector 391 corresponding to the wavelength dependent lateral aberrationat a field position corresponding to surface feature 386 is determinedby cross-correlating sub-regions 384, 386. The length of an arrow 393corresponds to a vector magnitude of one detector pixel. Vectorsdetermined from multiple pairs of corresponding sub-regions can bepresented as a vector map indicative of the field dependent wavelengthdependent lateral aberration of the interferometer.

While the forgoing method has been described as using an interferometerconfigured so that either wavelength or geometry dependent lateralaberration will dominate, other methods can be used. For example,wavelength dependent lateral aberration can be determined at highnumerical aperture by making multiple measurements at each of differentwavelengths (e.g., by narrowly filtering a broadband source under highnumerical aperture illumination conditions). Alternatively, wavelengthdependent lateral aberration can be determined in an interferometernominally configured for high numerical aperture illumination (e.g., byspatially filtering a broadband source to reduce geometrical lateralaberration as compared to wavelength dependent lateral aberration).

We next discuss exemplary applications of methods for determininginformation related to lateral aberration (e.g., a vector map ofwavelength dependent and/or geometry dependent lateral aberration).

In some embodiments, information related to lateral aberration is usedto position one or more optical components of an interferometer. Forexample, information related to lateral aberration can be used as amanufacturing feedback mechanism to assist determining the center, tilt,tip, and/or longitudinal spacing of an optical element that reduceslateral aberration. The positioning is typically based on informationrelated to lateral aberrations over the entire field of view of theinterferometer.

Typically, the method includes determining first information related tolateral aberration. For example, a first vector map can be preparedbased on interference signals obtained with optical elements of theinterferometer in a first position. The position (e.g., center, tilt,tip, and/or lateral spacing) of one or more of the optical elements(e.g., objective) is modified. A second vector map is prepared based oninterference signals obtained with the optical elements of theinterferometer in the modified position. The first and second vectormaps can be compared to determine whether the lateral aberrations werereduced or increased by the modified position of the optical element(s).For example, the magnitude of vectors within the vector maps can becompared. This process can continue iteratively until, for example,lateral aberrations have been reduced to less than a determined level.

As an example, consider that the camera lens of a microscope objectivecan be positioned over a range of distances from the pupil of themicroscope objective (e.g., the longitudinal spacing between the cameralens and pupil can vary (e.g., over tens of millimeters)). Suchmicroscope objectives can be positioned along an optical path of aninterferometer. Vector maps obtained at different longitudinal positionsof the camera lens and pupil can be used to reduce lateral aberrationthat results from improper longitudinal spacing between the camera lensand pupil. Such vector maps can also be used to determine, for example,a center, a tip, and a tilt of the lens that reduces lateral aberration.

Referring to FIG. 21, a 500 for determining a lateral aberration of anoptical element (e.g., a telecentric lens system 501) includes a lightsource 502, a beam splitter 503, a patterned test object 505, areference object 507 movable along a scan dimension 509, and amultidimensional detector 511. Light source 502 can be configured toilluminate object 505 with, for example, broadband light having a lownumerical aperture (e.g., using a spatial filter to restrict angles ofincidence) or narrow band light having a high numerical aperture (e.g.,using a spectral filter to restrict wavelengths of the illuminatinglight). The lateral aberration of lens system 501 can be determinedbased on, for example, information related to multiple spatial locationsof a patterned test object (e.g., an image, phase profile, and/or heightprofile of the test object) and cross-correlation of sub-regions of theinformation related to the spatial locations as discussed above. Forexample, a vector map of geometry and/or wavelength dependent lateralaberrations can be determined. Optical elements of lens system 501 canbe iteratively adjusted based on the information about the lateralaberration. Lateral aberration of lens system 501 can be compared tolateral aberration of reference lens.

Referring to FIG. 22, a system 515 for determining a lateral aberrationof an optical element (e.g., afocal lens system 516) includes a lightsource 502, a beam splitter 518, 25 patterned test object 505, referenceobject 507 movable along scan dimension 509, a pair of microscopeobjectives 517, 519, an optic 520 (e.g., an infinite conjugateobjective), and multidimensional detector 511. System 515 can be used assystem 500 (e.g., to iteratively adjust optical elements of lens system516 and/or compare lateral aberration of lens system 501 to lateralaberration of a reference lens).

While systems 500 and 515 have been described as being used to determinea lateral aberration of a particular optical element, such systems canbe used to determine a lateral aberration of other optical elements aswell. In general, an interferometer can be used to determine a lateralaberration of any optical element positioned along an optical path ofthe interferometer. The lateral aberration that is determined may be thecontribution of the optical element to the total lateral aberration ofall the optical elements of the interferometer.

While a method for determining information about lateral aberration hasbeen described as being based on light reflected from a patterned testobject, other test objects may be used. For example, the test object mayhave random surface features (e.g., surface roughness due to machiningand/or chemical etching). Correlation can be performed on sub-regionsdefined by, for example, a regular sampling array.

While a method for determining information about lateral aberration hasbeen described as being based on information about a test objectdetermined from each of two different frequencies, more frequencies ofthe interference signals can be used. For example, use of more than twofrequencies can provide information about the lateral aberrations over agreater range of wavelengths of the light source and angles ofincidence. Additionally, if more than two frequencies are used, thecorrelation can be performed using information about the test objectdetermined from more closely spaced frequencies. By using multiplecorrelation steps across most (e.g., all) of the range of frequenciescorresponding to the effective spatial frequency spectrum of theinterferometer it is possible to provide information between frequenciesthat are so widely separated that direct correlation would be poor, forexample because of the blurring of images at different colors.

As discussed above, information about the test object (e.g., areflectivity image, a phase profile, or a height profile) can bedetermined based on interference signals from a test object. Also asdiscussed above, information about the test object (e.g., a reflectivityimage, a phase profile, or a height profile) can be determined based oneach of multiple frequency components of the interference signals. Forexample, as discussed with respect to FIG. 19, height profiles of anobject can determined from each of different frequency components ofinterference signals.

The ITF of an interferometer is indicative of the response of theinterferometer to height variations of a test object along the scandimension. Typically, the height variations are expressed as a heightprofile of the test object. A height profile of a test object can bedetermined from, for example, phase information of interference signalsobtained from the test object. For example, FIGS. 19, 20A and 20Billustrate height profiles of sharp surface features of a test objectgenerated using phase information at each of two different frequencycomponents of multiple interference signals.

The ITF of an interferometer can be determined based upon a test objectheight profile determined based on interference signals obtained withthe interferometer. Typically, an ITF is determined by, for example,Fourier transforming a height profile and determining the ratio of thetransformed height profile spectrum and a reference Fourier transform(e.g., the Fourier transform of a reference surface feature (e.g., asurface feature having a step function in height)). An ITF can bedetermined for each of multiple positions within a field of view of theinterferometer (e.g., the ITF can be determined based on the heightprofiles of each of multiple sub-regions of a test object).

An ITF can be determined based upon each of multiple height profileswhere each height profile is determined from a different frequencycomponent of interference signals from a test object. Each ITF isindicative of the response of the interferometer to height variations ofthe test object at the wavelength of light that corresponds to thefrequency component used to prepare the height profile for that ITF.Hence, the multiple ITF's can be used to determine information relatedthe wavelength dependence of the ITF for the interferometer.

The MTF of an interferometer is indicative of the response of theinterferometer to reflectivity variations along a lateral dimension of atest object test. Typically, the reflectivity variations are expressedas an image of the test object.

The MTF of an interferometer can be determined based upon an image ofthe test object. Typically, an MTF is determined by, for example,Fourier transforming an image of a test object and determining the ratioof the transformed image and a reference Fourier transform (e.g., theFourier transform of a reference surface feature (e.g., a surfacefeature having sharp reflectivity transition)). An MTF can be determinedfor each of multiple positions within a field of view of theinterferometer (e.g., the MTF can be determined based on the images ofeach of multiple sub-regions of a test object).

An MTF can be determined based upon each of multiple images where eachimage is determined from a different frequency component of interferencesignals from a test object. Each MTF is indicative of the response ofthe interferometer to reflectivity variations of the test object at thewavelength of light that corresponds to the frequency component used toprepare the height profile for that MTF. Hence, the multiple MTF's canbe used to determine information related the wavelength dependence ofthe MTF for the interferometer.

The location and orientation of the discontinuous features of a testobject used to determine the ITF and/or MTF can be chosen to measurethese properties in various directions (vertically, horizontally,radially or tangentially for example), which allows studying thestructure of certain types of non-symmetrical aberrations.

Note that in some cases additional information may be gained by movingthe detector through focus and repeating the measurement.

While scanning interferometry data have been described as being obtainedby varying an OPD (e.g., by moving a test and/or reference object),other configurations are possible. For example, in some embodiments,scanning interferometry data are obtained by varying a wavelength ofthat light interferes at the detector. Each scan position typicallycorresponds to a different wavelength of detected interfering light(e.g., to a different central wavelength of the detected interferinglight). Each scan position increment typically corresponds to adifference in the wavelength between scan positions.

Any of the methods described above can be implemented, for example, incomputer hardware, software, or a combination of both. The methods canbe implemented in computer programs using standard programmingtechniques following the descriptions herein. Program code is applied toinput data to perform the functions described herein and generate outputinformation. The output information is applied to one or more outputdevices such as a display monitor. Each program may be implemented in ahigh level procedural or object oriented programming language tocommunicate with a computer system. However, the programs can beimplemented in assembly or machine language, if desired. In any case,the language can be a compiled or interpreted language. Moreover, theprogram can run on dedicated integrated circuits preprogrammed for thatpurpose.

Each such computer program is preferably stored on a storage medium ordevice (e.g., ROM or magnetic diskette) readable by a general or specialpurpose programmable computer, for configuring and operating thecomputer when the storage media or device is read by the computer toperform the procedures described herein. The computer program can alsoreside in cache or main memory during program execution. The analysismethod can also be implemented as a computer-readable storage medium,configured with a computer program, where the storage medium soconfigured causes a computer to operate in a specific and predefinedmanner to perform the functions described herein.

Other aspects, features, and advantages are within the scope of theinvention.

1. A method comprising: providing scanning interferometry data from aninterferometer, wherein: the interferometer comprises multiple opticalelements configured to reflect light from a test object different fromthe optical elements to determine information about the test object, andthe scanning interferometry data comprises an interference signalcomprising an interference intensity value for each of multiple scanpositions of the interferometer; determining a relationship betweenphase and frequency components of the interference signal; and reducing,based on the relationship between the phase and frequency components ofthe interference signal, an optical thickness mismatch for a fieldposition of the interferometer corresponding to the interference signal,the optical thickness mismatch being between the optical elements of areference arm of the interferometer and the optical elements of a testarm of the interferometer.